## JSH: So why tautological spaces?

Lost in all the arguing around this or that research of mine is the tool used to find mathematical solutions like the simplification to binary quadratic Diophantine equations which I call a tautological space.

At its simplest it's just an identity like: x+y+vz = x+y+vz, but written as x+y+vz = 0(mod x+y+vz), and I'd guess that few of you are into modular arithmetic or it's also called congruences, if you are physics people as I was not, but learned their utility.

Notice you have x, y and z, plus an extra variable I call v. (I picked v as the variable for various reasons.)

That extra variable v is your handle on the space. It can be ANY VALUE you wish! And I invented this approach because I was frustrated fiddling with an equation where I'd exhausted all possible ways I could think of, of manipulating x, y and z.

ALL the major controversy you see with Usenet posters who relentlessly reply to my postings is around the research given by tautological spaces. The paper that went to the math journal that published it, withdrew it, and then died, was from tautological spaces.

The postings I've done on binary quadratic Diophantine equations is with research I did using tautological spaces.

To me it's simple enough: math people will kill one of their own journals, stalk me relentlessly, and spread as much disinformation as possible to distract people from using this technique I call tautological spaces. So I understand what they're doing. While for the rest of you it's probably a mystery, as to what could be the big deal with such a thing?

Well tautological spaces again are just identities. Because they are identities the basic tautological space is always true, and you just take the identity, manipulate it algebraically a bit, and subtract ANY equation you wish, where I'd assume it'd have only x, y and z, and then you analyze the residue.

The residue then has all its properties from the equation you subtracted from it, which I call the conditional, but it also has this extra variable v, which is YOURS. You can make v whatever you wish, which means your creative mind can come into play and you can actually talk to the mathematics you might say, and at times I feel like it's more of a conversation with the math.

It's like you are partners.

The story here is way screwy as yes, math people already killed one of their own math journals fighting this research. And you can see the arguing over my latest use of it with binary quadratic Diophantine equations in recent threads all over the sci.physics newsgroup.

Oh, I'd like to leave you with one more thing, which I re-discovered because of these latest musings, and let you ponder why mathematicians would let this be lost, or mainly just not noted as important!

Given x^2 - Dy^2 = 1, in rationals:

y = 2t/(D - t^2)

and

x = (D + t^2)/(D - t^2)

and you get hyperbolas with D>0, the circle with D=-1, and ellipses in general with D<0.

I think that is beautiful and yet it was known to Fermat. Why isn't it taught today in even baby physics courses? The D is related to eccentricity. You can produce every ellipse or hyperbola—just a small, I guess unit-like version with ellipses—by just fiddling with that D number.

But you see, mathematicians don't see that as an equation for use with rationals as they call it "Pell's Equation" and only care about it as a Diophantine equation, that is, only for integers.

Math people are quirky. They can do the damndest things. So yeah, I invented this technique I call tautological spaces and in warring against it they've killed one of their OWN math journals, and you can see the continual stalking on the newsgroups.

Why? Ask them. I dare you. Just even the rational parameterization. Ask some math person why that is buried within the math literature rather than being trumpeted as an interesting way, at a minimum, to graph ellipses and hyperbolas.

And who knows? What if the D number said something about orbits that eccentricity does not, when you look at the data laid out?

What if our universe cares more about the D number than it does about eccentricity?

Can you say right now whether the physics looks different with one number or the other when you consider orbit classification?